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According to this question uniform convergence of characteristic functions put forward by Shine, we get following result.

Suppose that a sequence of probability measures $\mu_n$ on $(\mathbb{R},\mathfrak{B}_\mathbb{R})$ converges weakly to $\mu$. Let $\phi_n$ and $\phi$ denote respetively the characteristic function of $\mu_n$ and $\mu$. Then $\phi_n$ converges uniformly to $\phi$ on any compact subset $K\subseteq\mathbb{R}$.

It seems that the uniform convergence of $\phi_n$ to $\phi$ doesn't hold on the whole real line $\mathbb{R}$. Could we find some simple counterexamples? Equivalently, could we find a sequence of random variables $\{X_n\}_{n=1}^\infty$ such that $X_n\xrightarrow{D}X_\infty$ but $\{\phi_{X_n}(t)\}$ doesn't uniformly converge to $\phi_{X_\infty}(t)$ on $\mathbb{R}$?

Jay
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I think $X_n \equiv \frac{1}{n}$ works. They converge to $X_{\infty} \equiv 0$. But $\phi_{X_n}(t) = e^{\frac{it}{n}}$, and $\phi_{X_{\infty}} \equiv 1$. No matter how large $n$ is, we can pick $t$ (e.g. $t = n$) so that $\phi_{X_n}(t)$ is bounded away from $1$.

JH19880819
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